2026/07/27 by Shiyu Yan
Mathematics · #math.CO
The layer number of a finite point set is the number of iterations needed to delete it by repeatedly removing the vertices of its convex hull. Ambrus, Hsu, Peng, and Yan conjectured that the layer number of the d-dimensional integer grid \1,…,n\d is of order n2d/(d+1) for every fixed d. We prove this conjecture. Let Pi be the convex hull of the point set remaining after i steps, and let Zn be the convex hull of the lattice points in the Euclidean ball of radius n. For every step that leaves a nonempty point set, the Minkowski sum Pi+1+Zn contains no vertex of Pi+Zn. Integrality of normalized lattice volume, together with the Bárány--Larman vertex estimate for Zn, gives a lower bound, independent of i, on the resulting volume decrease. Summing over i yields the matching upper bound, even when Pi is lower-dimensional. For d≥2, the same upper bound holds uniformly over all nonempty subsets of \1,…,n\d.